3.1.11 \(\int \csc ^4(e+f x) (-3+2 \sin ^2(e+f x)) \, dx\) [11]

Optimal. Leaf size=18 \[ \frac {\cot (e+f x) \csc ^2(e+f x)}{f} \]

[Out]

cot(f*x+e)*csc(f*x+e)^2/f

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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {3090} \begin {gather*} \frac {\cot (e+f x) \csc ^2(e+f x)}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^4*(-3 + 2*Sin[e + f*x]^2),x]

[Out]

(Cot[e + f*x]*Csc[e + f*x]^2)/f

Rule 3090

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A*Cos[e
+ f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1))), x] /; FreeQ[{b, e, f, A, C, m}, x] && EqQ[A*(m + 2) + C*(m +
1), 0]

Rubi steps

\begin {align*} \int \csc ^4(e+f x) \left (-3+2 \sin ^2(e+f x)\right ) \, dx &=\frac {\cot (e+f x) \csc ^2(e+f x)}{f}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 18, normalized size = 1.00 \begin {gather*} \frac {\cot (e+f x) \csc ^2(e+f x)}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]^4*(-3 + 2*Sin[e + f*x]^2),x]

[Out]

(Cot[e + f*x]*Csc[e + f*x]^2)/f

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Maple [A]
time = 0.24, size = 34, normalized size = 1.89

method	result	size
derivativedivides	\(\frac {-3 \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )-2 \cot \left (f x +e \right )}{f}\)	\(34\)
default	\(\frac {-3 \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )-2 \cot \left (f x +e \right )}{f}\)	\(34\)
risch	\(-\frac {4 i \left ({\mathrm e}^{4 i \left (f x +e \right )}+{\mathrm e}^{2 i \left (f x +e \right )}\right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3}}\)	\(39\)
norman	\(\frac {\frac {1}{8 f}+\frac {3 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}+\frac {\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}-\frac {\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}-\frac {3 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}\)	\(114\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)^4*(-3+2*sin(f*x+e)^2),x,method=_RETURNVERBOSE)

[Out]

1/f*(-3*(-2/3-1/3*csc(f*x+e)^2)*cot(f*x+e)-2*cot(f*x+e))

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Maxima [A]
time = 0.27, size = 24, normalized size = 1.33 \begin {gather*} \frac {\tan \left (f x + e\right )^{2} + 1}{f \tan \left (f x + e\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^4*(-3+2*sin(f*x+e)^2),x, algorithm="maxima")

[Out]

(tan(f*x + e)^2 + 1)/(f*tan(f*x + e)^3)

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Fricas [A]
time = 0.52, size = 35, normalized size = 1.94 \begin {gather*} -\frac {\cos \left (f x + e\right )}{{\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^4*(-3+2*sin(f*x+e)^2),x, algorithm="fricas")

[Out]

-cos(f*x + e)/((f*cos(f*x + e)^2 - f)*sin(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (2 \sin ^{2}{\left (e + f x \right )} - 3\right ) \csc ^{4}{\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)**4*(-3+2*sin(f*x+e)**2),x)

[Out]

Integral((2*sin(e + f*x)**2 - 3)*csc(e + f*x)**4, x)

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Giac [A]
time = 0.60, size = 24, normalized size = 1.33 \begin {gather*} \frac {\tan \left (f x + e\right )^{2} + 1}{f \tan \left (f x + e\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^4*(-3+2*sin(f*x+e)^2),x, algorithm="giac")

[Out]

(tan(f*x + e)^2 + 1)/(f*tan(f*x + e)^3)

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Mupad [B]
time = 13.17, size = 18, normalized size = 1.00 \begin {gather*} \frac {\cos \left (e+f\,x\right )}{f\,{\sin \left (e+f\,x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*sin(e + f*x)^2 - 3)/sin(e + f*x)^4,x)

[Out]

cos(e + f*x)/(f*sin(e + f*x)^3)

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